ENGINEERING DRAWING (CHAPTER -6)
๐ ENGINEERING DRAWING
Chapter 6 – Engineering Curves
6.1 Introduction
In engineering, certain special curves frequently appear in the design of gears, cams, springs, bridges, trusses, aeroplane wings, turbines, and machine parts.
These are not just mathematical curves but have direct engineering applications.
The main engineering curves are:
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Conic Sections – Ellipse, Parabola, Hyperbola
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Cycloidal Curves – Cycloid, Epicycloid, Hypocycloid
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Involute
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Spirals – Archimedean Spiral, Logarithmic Spiral
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Helix – Cylindrical Helix, Conical Helix
6.2 Conic Sections
A conic is obtained when a right circular cone is cut by a plane at different inclinations.
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Ellipse – Obtained when plane cuts the cone at an angle smaller than side of cone.
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Parabola – Obtained when plane is parallel to the generator of cone.
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Hyperbola – Obtained when plane cuts both halves of the cone.
(A) Ellipse
Definition: The locus of a point moving in a plane such that the sum of its distances from two fixed points (foci) is constant.
Methods of Construction:
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Concentric Circle Method
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Draw two concentric circles with major radius and minor radius.
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Divide both into equal parts.
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Project points radially and join → ellipse.
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Rectangle Method
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Draw a rectangle equal to major × minor axis.
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Divide sides into equal parts.
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Project points and join smoothly.
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Applications:
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Gear tooth profiles, cams, railway arches, aeroplane windows.
(B) Parabola
Definition: The locus of a point moving in such a way that its distance from a fixed point (focus) equals its distance from a fixed line (directrix).
Construction (Rectangle Method):
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Draw a rectangle with base = axis length, height = ordinate.
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Divide base into equal parts.
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Draw verticals from divisions, horizontals from ordinate divisions.
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Intersections → parabola.
Applications:
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Reflectors of car headlights, satellite dishes, bridges, motion of a projectile.
(C) Hyperbola
Definition: The locus of a point moving such that the difference of its distances from two fixed points (foci) is constant.
Construction (Rectangle / Asymptote Method):
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Draw transverse and conjugate axes.
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Draw asymptotes at right angles.
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Locate points by intersecting rectangular ordinates.
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Join smoothly.
Applications:
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Cooling towers, navigation systems, optics (radar, radio location).
6.3 Cycloidal Curves
These are generated by a point on the circumference of a circle rolling on a line or another circle.
(A) Cycloid
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Curve traced by a point on the rim of a circle rolling along a straight line (without slipping).
Construction:
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Draw a line (base).
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Roll circle along line for one revolution.
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Divide circle and base into equal parts.
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Trace locus of rim point.
Applications:
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Tooth profiles of gears, roller paths, railway wheels.
(B) Epicycloid
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Generated when a circle rolls outside another circle.
Applications:
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Gear tooth profiles (external meshing).
(C) Hypocycloid
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Generated when a circle rolls inside another circle.
Applications:
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Gear profiles (internal meshing), cams.
6.4 Involute
Definition: The locus of a point on a string as it is unwound from a circle (or polygon).
Construction:
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Draw a base circle.
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Divide circumference into equal parts.
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At each division, draw tangents.
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Mark length of arc on tangent.
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Connect points smoothly → involute.
Applications:
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Gear tooth profile (standard involute gears).
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Pulley and belt design.
6.5 Spirals
(A) Archimedean Spiral
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A curve generated by a point moving around a fixed center with uniform angular velocity and uniform linear velocity away from the center.
Applications:
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Scroll compressors, cam profiles, gramophone records.
(B) Logarithmic Spiral
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A curve where the angle between radius vector and tangent is constant.
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Radius increases exponentially.
Applications:
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Antennas, propeller blades, turbine blades, seashells in nature.
6.6 Helix
A curve traced by a point moving around a cylinder with uniform angular velocity and uniform linear velocity along its axis.
Types:
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Cylindrical Helix – Constant radius.
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Conical Helix – Radius decreases or increases gradually.
Applications:
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Springs (compression, tension), helical gears, threads of screws, drills.
6.7 Summary of Chapter
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Conics: Ellipse (sum of distances = constant), Parabola (focus = directrix), Hyperbola (difference of distances = constant).
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Cycloidal curves: Cycloid, Epicycloid, Hypocycloid – rolling circle curves.
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Involute: Generated by unwinding a string – used in gear teeth.
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Spirals: Archimedean (linear increase), Logarithmic (exponential increase).
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Helix: Generated on a cylinder/cone – used in springs, screws, drills.
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